The derivation on the diagram is based on mathematical induction, which can also be proved by Taylor formula and general formula, but it is rather puzzling! Binomial theorem is widely used in combinatorial theory, higher power, higher arithmetic progression summation and difference method. For example: formula 1, CN0+cn1+cn2 …+cnk+…+CNN = 2n, formula 2, cno-cn1+cn2-cn3+… (-1) ncnn = 0. If a=b= 1, the first formula can be proved, and if a=- 1 and b= 1, the second formula can be proved. Formula plus formula 2 gives: 2(cn0+cn2+cn4+……)= 2n to get cn0+cn2+cn4+… = 2 (n-1); Subtract Formula 2 from Formula 1: 2 (cn1+cn3+cn5+) ...) = 2n; So what? cn 1+cn3+cn5+……=2^(n- 1)； Do you know? Cn0+cn2+cn4+... = cn1+cn3+cn5+... = 2 (n-1), so the third formula is proved.